Forced Response Amplitudes Research Articles

An Efficient Approach for Determining Forced Vibration Response Amplitudes of a MDOF System with Various Attachments

The frequency-response curve is an important information for the structural design, but the conventional time-history method for obtaining the frequency-response curve of a multi-degree-of-freedom (MDOF) system is time-consuming. Thus, this paper presents an efficient technique to determine the forced vibration response amplitudes of a multi-span beam carrying arbitrary concentrated elements. To this end, the "steady" response amplitudes|Y(x)|sof the above-mentioned MDOF system due to harmonic excitations (with the specified frequencieswe) are determined by using the numerical assembly method (NAM). Next, the corresponding "total" response amplitudes|Y(x)|tof the same vibrating system are calculated by using a relationship between|Y(x)|tand|Y(x)|sobtained from the single-degree-of-freedom (SDOF) vibrating system. It is noted that, near resonance (i.e.,we/w≈1.0), the entire MDOF system (with natural frequencyw) will vibrate synchronously in a certain mode and can be modeled by a SDOF system. Finally, the conventional finite element method (FEM) incorporated with the Newmark's direct integration method is also used to determine the "total" response amplitudes|Y(x)|tof the same forced vibrating system from the time histories of dynamic responses at each specified exciting frequencywe. It has been found that the numerical results of the presented approach are in good agreement with those of FEM, this confirms the reliability of the presented theory. Because the CPU time required by the presented approach is less than 1% of that required by the conventional FEM, the presented approach should be an efficient technique for the title problem.

Interaction of Waves With a Steady Intake/Discharge Flow Emanating From a 3D Body

It has been proposed that the warm surface-water intake pipes distributed around an OTEC plant can generate adequate momentum to globally position a platform to overcome the second-order drift forces, thereby eliminating the need for additional power for thrusters or for mooring lines. It is evident that if the intake rate of the flow is high, there will be interaction among the locally created steady flow due to the intake, the incoming wave, and the ensuing platform motions. In this work, we address such concerns by developing a linear theory for obtaining the motions (in the presence of incoming waves) of arbitrary 3D bodies from which there is a steady intake/discharge. The boundary-value problem is formulated within the assumption of the linear potential theory by decomposing the total potential into oscillatory and steady components. The steady potential is further decomposed into double-model and perturbation potentials. The time harmonic potential is coupled with the steady potential through the free-surface condition. The potentials are obtained using the quadratic boundary-element method. The effect of the steady flow on hydrodynamic force coefficients and response amplitude operators is studied.

Dissipated energy and boundary condition effects associated to dry friction on the dynamics of vibrating structures

In turbomachines, dry friction devices (under platform dampers, shrouds, and tie-wire) are usually introduced to reduce resonant responses of bladed disks. Dry friction between rubbing elements induces a highly nonlinear dynamic behaviour which flattens the frequency response functions. It is clear that such behaviour requires an optimisation process to find the optimum parameters that lead to the minimum forced response amplitudes. However, different interpretations still remain concerning the explanation of the physical origin of this type of flattening. The most common one is based on dissipated energy. In this case, heat resulting from the relative frictional motion between contacting surfaces is supposed to bring sufficient dissipation to flatten response functions. On the other hand, a different approach considers that a decrease in vibrational amplitudes is explained by changes in boundary conditions induced by a stick/slip behaviour. In this study, a single degree-of-freedom system is used and analysed both in time and in frequency domains (Harmonic Balance Method) in order to show the contribution of respectively energy dissipation and change of contact state on peak levels.

Effects of Additional Mass on Modal Experimental Analysis of 3-Dimension and 5-Direction Braided Composites

The work of vibration test has significant meanings for the researches and applications of 3-dimension and 5-direction braided composites. This article discusses the effects of added mass with different weight on the modal test of 3-dimension and 5-direction braided composites. The comparison of the modal parameters of 3-dimension and 5-direction braided composites tested by different weight of mass reveals that the additional mass is a mostly influence factor for vibration property of 3-dimension and 5-direction braided composites. The results of frequency response and force response curves show that smaller mass accelerometer is more effective for a wider range of frequencies around the resonance frequency, a higher natural frequency and a larger peak in these points. Force-response curves show that force response amplitude increases with the increase of additional mass weight, and the larger additional mass, the shorter time taken for reaching stationary state. The errors of natural frequency and damping ratio increase when the weight of additional mass increases. With the increase of modal orders, relative errors of modal characteristics have slighter decreasing degrees. The results derived from this article will provide a useful reference for precise modal analysis of 3-dimension and 5-direction braided composites.

Effect of Bimodularity on Dynamic Response of Cross-ply Conical Panels

The effect of bimodularity ratio on free vibration characteristics of cross-ply conical panels of various geometry and lamination scheme is studied. Forced vibration response is also studied for a typical case. The formulation is based on first order shear deformation theory and Bert’s constitutive model. An iterative eigenvalue approach is employed to obtain the positive and negative half cycle free vibration frequencies. Galerkin’s approach in time domain is used to obtain the frequency response. It is interesting to note that there is a significant difference between positive and negative half cycle frequencies depending on panel parameters. Also, there is a significant difference in positive and negative half cycle forced response amplitudes due to bimodularity. Bimodularity, the different behavior of material in tension and compression, affects the static and dynamic response of structures. Few studies on static analysis of bimodulus laminated cross-ply composite shells and dynamic analysis of cross-ply panels have been presented [1-4]. To the best of the authors’ knowledge, the work on the analysis of bimodular laminated cross-ply conical shell panels is not dealt in the literature. The effect of bimodularity ratio on free and forced vibration characteristics of cross-ply conical panel is important for design of such structures under dynamic loading condition. Here, the dynamic analysis of cross-ply laminated conical panels of bimodulus material is carried out using finite element method and Bert’s constitutive model. The effect of semi-cone angle, number of layers and bimodularity ratio (E 2t /E 2c ) on free vibration frequencies and frequency response is investigated. 2. FORMULATION The geometry and dimensions of a conical panel is shown in Fig. 1 with total thickness h, small end radius r 1 , large end radius r 2 , meridional length L, circumferential length b at small end, sector angle y, and semi-cone angle a. Displacements u, v, w at a point (s, q, z) are expressed as functions of middle surface displacements u 0 , v 0 , w 0 and independent rotation b s , b q of the meridional and hoop sections, respectively, as: u (s, q, z, t) = u 0 (s, q, t) + z b s (s, q, t) v (s, q, z, t) = v 0 (s, q, t) + z b q (s, q, t) (1) w (s, q, z, t) = w 0 (s, q, t)

Static Stress Redesign of Plates by Large Admissible Perturbations

The purpose of this paper is to develop further the large admissible perturbation (LEAP) methodology to solve the static stress redesign problem for shell elements. The static stress general perturbation equation, which expresses the unknown stresses of the objective structure in terms of the baseline structure stresses, is derived first. This equation depends on the redesign variables for each element or group of elements, namely, the plate thickness. LEAP enables the designer to redesign a structure to achieve specifications on modal properties, static displacements, forced response amplitudes, and static stresses. LEAP is implemented in code RESTRUCT, which postprocesses the finite element analysis FEA results of the baseline structure. Changes on the order of 100% in the above performance particulars and in redesign variables can be achieved without repetitive FEAs. Several numerical applications on a simple plate and an offshore tower are used to verify the effectiveness of the LEAP algorithm for stress redesign.

Reducing MRI gradient coil vibration with rib stiffeners

AbstractThis work investigates the effect of rib stiffeners on the free and forced vibration of a gradient coil in a magnetic resonance imaging (MRI) scanner. Several reinforcement schemes are studied in this article. One scheme utilizes the existing holes in the gradient coil structure (typically reserved for magnetic shims) to produce the reinforcement. Nonferrous, nonmagnetic carbon fiber rib stiffeners are employed to fill these holes in several ways to strengthen a gradient coil. Another scheme replaces the inner half of the gradient coil material with a grid of interconnected axial and circumferential rib stiffeners. It is found that the structural stiffness of the gradient coil increases substantially when the coil is reinforced by carbon fiber rib stiffeners. The reinforcement affects the noise and vibration response of the gradient coil structure in the following ways. It increases the frequency range of forced response of the gradient coil at low frequencies due to the increased resonant frequency of the fundamental mode of the coil. Second, it reduces the forced response amplitude of the coil structure (which is governed by the structural stiffness of the coil). Thirdly, it reduces the number of natural modes in the low and medium frequency range and therefore lessens the chance of the coil structure being excited resonantly by magnetic resonance signal acquisition sequences. It is shown that gradient coils modeled by solid finite element models have higher stiffness along the coil's circumference and lower stiffness in the axial direction than those using shell finite element models. © 2009 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 35B: 198–209, 2009

Minimizing Maximum Modal Force in Mistuned Bladed Disk Forced Response

Amplitude magnification is defined as the maximum forced response amplitude of any blade on a mistuned bladed disk divided by the maximum response amplitude of any blade on a tuned bladed disk over a range of engine order excitation frequencies. This paper shows that amplitude magnification can be approximated as the maximum ratio of modal force divided by modal vector magnitude in an isolated family of turbine engine bladed disk modes. An infinite linear mistuning pattern, defined by a constant interblade stiffness increment between an infinite number of blades, is found to minimize the maximum modal force when subjected to engine order N/4 excitation. Linear mistuning, an approximation of the infinite linear mistuning pattern, approximately minimizes the maximum modal force for bladed disks with a finite number of blades when subjected to engine order N/4 excitation. From this theory, 2/N is proposed to be a lower boundary for amplitude magnification. The linear mistuning method is demonstrated to produce very low amplitude magnifications numerically and experimentally. The numerical examples suggest that linear mistuning may produce amplitude magnifications near the absolute minimum possible.

Topology redesign for performance by large admissible perturbations

A methodology is developed for optimal structural topology design subject to several performance constraints. Eight-node solid elements are used to model the initial structure, which is a uniform solid block satisfying the boundary conditions and subjected to external loading. The Young modulus of each solid element or group of elements is used as redesign variable. A minimum change function is used as an optimality criterion. Performance constraints include static displacements, natural frequencies, forced response amplitudes, and static stresses. These constraints are treated by the large admissible perturbation methodology which makes it possible to achieve the performance objectives incrementally without trial and error or repetitive finite element analyses for changes in the order of 100–300%. Thus, the optimal topology is reached in about four to five iterations, where each iteration includes one finite element analysis and setting of an upper limit for the value of the modulus of elasticity to produce a manufacturable structure. Several numerical applications are presented using three different benchmark structures to demonstrate the methodology and the impact of performance constraints on the generated topology.

Integrated Redesign of Large-Scale Structures by Large Admissible Perturbations

In structural redesign (inverse design), selection of the number and type of performance constraints is a major challenge. This issue is directly related to the computational effort and, most importantly, to the success of the optimization solver in finding a solution. These issues are the focus of this paper, which provides and discusses techniques that can help designers formulate a well-posed integrated complex redesign problem. LargE Admissible Perturbations (LEAP) is a general methodology, which solves redesign problems of complex structures with, among others, free vibration, static deformation, and forced response amplitude constraints. The existing algorithm, referred to as the Incremental Method is improved in this paper for problems with static and forced response amplitude constraints. This new algorithm, referred to as the Direct Method, offers comparable level of accuracy for less computational time and provides robustness in solving large-scale redesign problems in the presence of damping, nonstructural mass, and fluid-structure interaction effects. Common redesign problems include several natural frequency constraints and forced response amplitude constraints at various frequencies of excitation. Several locations on the structure and degrees of freedom can be constrained simultaneously. The designer must exercise judgment and physical intuition to limit the number of constraints and consequently the computational time. Strategies and guidelines are discussed. Such techniques are presented and applied to a 2,694 degree of freedom offshore tower.

Static Stress Redesign of Structures by Large Admissible Perturbations

The LargeE Admissible Perturbation (LEAP) methodology is developed further to solve static stress redesign problems. The static stress general perturbation equation, which expresses the unknown nodal stresses of the objective structure in terms of the baseline structure stresses, is derived first. This equation depends on the redesign variables for each element or group of elements; namely, the cross-sectional area and moment of inertia, and the distance between the neutral axis and the outer fiber of the cross section. This equation preserves the shape of the cross section in the redesign process. LEAP enables the designer to redesign a structure to achieve specifications on modal properties, static displacements, forced response amplitudes, and static stresses. LEAP is implemented in code RESTRUCT which post-processes the FEA results of the baseline structure. Changes on the order of 100% in the above performance particulars and in redesign variables can be achieved without repetitive finite element (FE) analyses. Several numerical applications on a simple cantilever beam and an offshore tower are used to verify the LEAP algorithm for stress redesign.

Cognate Space Identification for Forced Response Structural Redesign

Large admissible perturbations (LEAP) is a general methodology, which solves redesign problems of complex structures with, among others, forced response amplitude constraints. In previous work, two LEAP algorithms, namely the incremental method (IM) and the direct method (DM), were developed. A powerful feature of LEAP is the general perturbation equations derived in terms of normal modes, the selection of which is a determinant factor for a successful redesign. The normal modes of a structure may be categorized as stretching, bending, torsional, and mixed modes and grouped into cognate spaces. In the context of redesign by LEAP, the physical interpretation of a mode-to-response cognate space lies in the fact that a mode from one space barely affects change in a mode from another space. Perturbation equations require computation of many perturbation terms corresponding to individual modes. Identifying modes with negligible contribution to the change in forced response amplitude eliminates a priori computation of numerous perturbation terms. Two methods of determining mode-to-response cognate spaces, one for IM and one for DM, are presented and compared. Trade-off between computational time and accuracy is assessed in order to provide practical guidelines to the designer. The developed LEAP redesign algorithms are applied to the redesign of a simple cantilever beam and a complex offshore tower.

Dynamic Response Predictions for a Mistuned Industrial Turbomachinery Rotor Using Reduced-Order Modeling

Abstract This paper explores the effects of random blade mistuning on the dynamics of an advanced industrial compressor rotor, using a component-mode-based reduced-order model formulation for tuned and mistuned bladed disks. The technique uses modal data obtained from finite element models to create computationally inexpensive models of mistuned bladed disks in a systematic manner. Both free and forced responses of the rotor are considered, and the obtained results are compared with “benchmark” finite element solutions. A brief statistical study is presented, in which Weibull distributions are shown to yield reliable estimates of forced response statistics. Moreover, a simple method is presented for computing natural frequencies of noninteger harmonics, using conventional cyclic symmetry finite element analysis. This procedure enables quantification of frequency veering data relevant to the assessment of mistuning sensitivity (e.g., veering curvatures), and it may provide a tool for quantifying structural interblade coupling in finite element rotor models of arbitrary complexity and size. The mistuned forced response amplitudes and stresses are found to vary considerably with mistuning strength and the degree of structural coupling between the blades. In general, this work demonstrates how reduced order modeling and Weibull estimates of the forced response statistics combine to facilitate thorough investigations of the mistuning sensitivity of industrial turbomachinery rotors.

Experimental Investigation of Mode Localization and Forced Response Amplitude Magnification for a Mistuned Bladed Disk

The results of an experimental investigation on the effects of random blade mistuning on the forced dynamic response of bladed disks are reported. The primary aim of the experiment is to gain understanding of the phenomena of mode localization and forced response blade amplitude magnification in bladed disks. A stationary, nominally periodic, 12-bladed disk with simple geometry is subjected to a traveling-wave out-of-plane “engine order” excitation delivered via phase-shifted control signals sent to piezoelectric actuators mounted on the blades. The bladed disk is then mistuned by the addition of small, unequal weights to the blade tips, and it is again subjected to a traveling wave excitation. The experimental data is used to verify analytical predictions about the occurrence of localized mode shapes, increases in forced response amplitude, and changes in resonant frequency due to the presence of mistuning. Very good agreement between experimental measurements and finite element analysis is obtained. The out-of-plane response is compared and contrasted with the previously reported in-plane mode localization behavior of the same test specimen. This work also represents an important extension of previous experimental study by investigating a frequency regime in which modal density is lower but disk-blade interaction is significantly greater.

Structural Redesign for Forced Response with Proportional Damping by Large Admissible Perturbations

A method to solve redesign (inverse design) problems of complex structures with forced response amplitude constraints is developed. The assumption is made that a structure is excited by harmonic external forces at a given frequency. The problem is to find optimum values of structural characteristics in order to achieve a desired level of forced response at one or several locations on the structure. The method of large admissible perturbations (LEAP) is used. The main advantage of LEAP is that a solution of the inverse problem can be found automatically without trial and error or repetitive finite element analyses. Using this method, algorithms with modal dynamic constraints and static displacement constraints have been developed and tested in the past. In this paper the redesign problem with forced response amplitude constraints is formulated. The method consists of two distinctive parts. First, the general perturbation equations are derived. They provide relations between the original structure S1, which is known and has undesired forced response amplitudes, and the unknown objective structure S2. Second, the redesign problem is solved by an incremental prediction-correction scheme, which permits large changes in redesign. Structural damping is considered in the form of Rayleigh damping. Under this formulation the damping matrix can be diagonalized by use of the real mode shapes of the undamped structure, which leads to the derivation of an exact perturbation equation with no loss of accuracy. Modal dynamic and static constraints may also be imposed simultaneously by the designer. The algorithm produces accurate results for large changes in response amplitude without additional finite element analyses. The number of extracted modes and the increment size control the accuracy of the results and the computational time. The importance of the choice of the objective function is discussed, and examples are presented.

Alterations of cross-bridge kinetics in human atrial and ventricular myocardium

We analyzed actomyosin cross-bridge kinetics in human atrial and ventricular muscle strip preparations by using sinusoidal length changes from 0.1 to 60 Hz. The minimum stiffness frequency was higher in atrial than in ventricular human myocardium and lower in failing than in non-failing left ventricular human myocardium. beta-Adrenergic stimulation increased the minimum stiffness frequency by 18 +/- 3% (p < 0.05). Cross-bridge kinetics are temperature-dependent, with a Q10 of at least 2.7. Dynamic stiffness measurements have revealed acute and chronic alterations of actomyosin cross-bridge kinetics in cardiac muscles of a variety of different animal species. We studied dynamic stiffness in right atrial and left ventricular preparations of non-failing and failing human hearts and tested the influence of the temperature and beta-adrenergic stimulation on cross-bridge kinetics. Muscle strips were prepared from right atria and left ventricles from human non-failing and failing hearts. After withdrawal of calcium, steady contracture tension was induced by the addition of 1.5 mM barium chloride. Sinusoidal length oscillations of 1% muscle length were applied, with a frequency spectrum of between 0.1 and 60 Hz. Dynamic stiffness was calculated from the length change and the corresponding force response amplitude. The specific minimum stiffness frequency, which indicates the interaction between cross-bridge recruitment and cross-bridge cycling dynamics, was analyzed for each condition: (1) The minimum stiffness frequency was 0.78 +/- 0.04 Hz in left ventricular myocardium and 2.80 +/- 0.31 Hz in right atrial myocardium (p < 0.01) at 27 degrees C. (2) The minimum stiffness frequency was 41% higher in non-failing compared to failing left ventricular human myocardium. (3) Over a wide range of experimental temperatures, the minimum stiffness frequency changed, with a Q10 of at least 2.7. (4) beta-Adrenergic stimulation significantly (p < 0.05) increased the minimum stiffness to 18 +/- 3% higher frequencies and significantly (p < 0.05) lowered contracture tension by 7 +/- 1%. The contractility of human heart muscle is not only regulated by excitation-contraction coupling but also by modulation of intrinsic properties of the actomyosin system. Acute and chronic alterations of cross-bridge kinetics have been demonstrated, which play a significant role in the physiology and pathophysiology of the human heart.

FREE AND FORCED RESPONSE OF MISTUNED LINEAR CYCLIC SYSTEMS: A SINGULAR PERTURBATION APPROACH

The effect of small deterministic parameter perturbations on the forced response of nearly periodic structures with cyclic symmetry is investigated. The general methodology developed herein is applicable to anyn-degree-of-freedom, strongly coupled cyclic system with two arbitrary and independent variations in system parameters that destroy the cyclic symmetry. The specific system studied may be regarded as a simplified model of a strongly coupled bladed disk assembly. Singular perturbation methods along with modal expansion analysis are applied to gain a physical insight into the effects of parameter perturbations on the eigenvalues, the eigenvectors as well as the forced response amplitudes. The study shows that, under appropriate conditions, the splitting and veering of eigenvalues due to mistunings or small parameter variations increases the amplitude of vibration of some blades significantly compared to what would be predicted by an analysis of the perfectly tuned system. Furthermore, modal bifurcations lead to uneven vibration amplitudes irrespective of the stiffness of the coupling springs. The variations in blade amplitudes are also found to be strongly dependent on damping, and the type of engine order excitation applied to the system for the same set of mistuning parameters.

Analysis of mistuned cyclic systems using mistune modes

The bladed disk assembly, a prime example of a cyclic system, forms an important component of modern turbomachinery. Vibrations of the blades leads to fatigue damage and limits their service life. Analysis of these systems is complicated by the combination of coupling of blade dynamics with mistuning of the blade properties. The present study of mistuned cyclic systems encompasses a complete model of a bladed disk assembly which includes multiple degrees-of-freedom per blade, aerodynamic coupling, and structural coupling. A systematic analysis of this model allows the determination of the effect of mistune pattern, along with excitation mode, on the forced response amplitudes of the blades. The main motivation of this study is to gain a better understanding of the dynamics of cyclic systems. Through this understanding, better designs of these components will be possible. The concept of mistuning modes is introduced to provide a generalized method of describing the mistuning of bladed disk assemblies. Perturbation analyses reveal which constant interblade phase angle modes will contribute to the response as a result of given mistune modes and excitation modes. Results generated through a computational study demonstrate the range of validity of the perturbation approaches.

Cross-bridge dynamics in human ventricular myocardium. Regulation of contractility in the failing heart.

To investigate whether altered cross-bridge kinetics contribute to the contractile abnormalities observed in heart failure, we determined the mechanical properties of cardiac muscles from control and myopathic hearts. Muscle fibers from normal (n = 5) and dilated cardiomyopathy (n = 6) hearts were obtained and chemically skinned with saponin. The muscles were then maximally activated at saturating calcium concentrations. Unloaded shortening velocities (V0) were determined in both groups. V0 in control was 0.72 +/- 0.09 Lmax/sec, whereas in myopathic muscles, V0 was 0.41 +/- 0.06 Lmax/sec at 22 degrees C. The muscles were also sinusoidally oscillated at frequencies ranging between 0.01 and 100 Hz. The dynamic stiffness of the muscles was calculated from the ratio of force response amplitude to length oscillation amplitude. At low frequencies (< 0.2 Hz) the stiffness was constant but was larger in myopathic muscles. In the range of 0.2-1 Hz, there was a drop in the magnitude of dynamic stiffness to approximately one quarter of the low-frequency baseline. This range reflects cross-bridge turnover kinetics. In control muscles, the frequency of minimum stiffness was 0.78 +/- 0.06 Hz, whereas it was 0.42 +/- 0.07 Hz in myopathic muscles. At higher frequencies, the dynamic stiffness increased and reached a plateau at 30 Hz. There were no differences in the plateau reached between control and myopathic muscles. Because myopathic hearts have a markedly diminished energy reserve, the slowing of the cross-bridge cycling rate plays an important adaptational role in the observed contractility changes in human heart failure. Although the potential to generate maximal Ca(2+)-activated force is similar in normal and myopathic hearts, alterations in contractile protein composition could explain the diminished cross-bridge cycling rate in failing hearts.

Dynamic stiffness measured in central segment of excised rabbit papillary muscles during barium contracture.

The dynamic mechanical behavior of excised rabbit papillary muscles that had been tonically activated by replacing bathing Ca2+ with Ba2+ was studied. Steady activation was used to visualize the dynamic behavior of cardiac myofilaments more clearly than is possible during twitches, which are complicated by the kinetics of excitation-contraction coupling. To avoid artifacts due to damaged ends of the muscle, the length of a central segment, which was defined by 2 tungsten pins inserted through the muscle, was measured. To test the mechanical behavior of the contractured muscles (at 24 degrees C), the central segment length was sinusoidally oscillated (amplitude 1%) at 15 different frequencies (0.05-30 Hz). The dynamic stiffness of the central segment was calculated from the ratio of force response amplitude to length perturbation amplitude. At low frequencies (below 0.4 Hz), stiffness was approximately constant and reflected the force-length relation. However, in a localized range near 1 Hz, there was a distinct drop in the magnitude of dynamic stiffness to approximately half its low-frequency baseline. This range may reflect the dynamics of attachment and detachment of force generators. The frequency of minimum stiffness was consistent among all muscles (1.3 +/- 0.3 Hz). Moreover, no significant change in this frequency was found over the examined range of lengths (90-100% of the segment length that produced maximal developed force) and activation levels (Ba2+ concentration 0.3-1.0 mM). From 2 to 8 Hz, dynamic stiffness appeared to reflect force-velocity properties, but at higher frequencies, another elastic property emerged. At 30 Hz, stiffness was proportional to force, with an apparent series elasticity less than 1.8%. Even though the muscles had only moderate longitudinal inhomogeneity, quantitatively significant (35%) errors would have been introduced had the study relied on total muscle length instead of central segment length.